Nonlinear Perron-Frobenius Theory and Dynamics of Cone Maps

Lemmens, Bas

doi:10.1007/3-540-34774-7_51

Cited by 6 publications

(7 citation statements)

References 15 publications

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“…. r) for some 1 ≤ r ≤ N. Then the above theorem reduces to a particular case of a result of Lemmens as stated in [5]. We now state the same as a corollary.…”

Section: A Generalisation Of Theorem (11) For Pairwise Commuting Matr...mentioning

confidence: 70%

Dynamics of products of nonnegative matrices

Jayaraman¹,

Kumar²,

Sridharan³

2020

Preprint

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The aim of this manuscript is to understand the dynamics of products of nonnegative matrices. We extend a well known consequence of the Perron-Frobenius theorem on the periodic points of a nonnegative matrix to products of finitely many nonnegative matrices associated to a word and later to products of nonnegative matrices associated to a word, possibly of infinite length. We also make use of an appropriate definition of the exponential map and the logarithm map on the positive orthant of R n and explore the relationship between the periodic points of certain subhomogeneous maps defined through the above functions and the periodic points of matrix products, mentioned above.

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“…. r) for some 1 ≤ r ≤ N. Then the above theorem reduces to a particular case of a result of Lemmens as stated in [5]. We now state the same as a corollary.…”

Section: A Generalisation Of Theorem (11) For Pairwise Commuting Matr...mentioning

confidence: 70%

Dynamics of products of nonnegative matrices

Jayaraman¹,

Kumar²,

Sridharan³

2020

Preprint

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“…For positive maps which are also order-preserving and sub-homogeneous, existing results (see next section for insights) do not provide any condition to ensure convergence to a fixed point, but only to periodic points [13] when the initial state is strictly positive, i.e., x ∈ R n + . Furthermore, to the best of our knowledge, no result provides any information about trajectories whose initial state lies in the boundary of R n ≥0 .…”

Section: Resultsmentioning

confidence: 99%

“…An extensive overview of these results was given by Hirsch and Smith [9]. On the other hand, in nonlinear Perron-Frobenius theory one usually considers discrete-time dynamical systems that need not be strongly order-preserving, but satisfy an additional concave assumption and obtain similar results regarding periodic trajectories [13]. The concave assumption of interest in this paper is sub-homogeneity.…”

Section: Sub-homogeneous Mapsmentioning

confidence: 96%

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A Nonlinear Perron-Frobenius Approach for Stability and Consensus of Discrete-Time Multi-Agent Systems

Deplano¹,

Franceschelli²,

Giua³

2019

Preprint

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In this paper we propose a novel method to establish stability and, in addition, convergence to a consensus state for a class of discrete-time Multi-Agent System (MAS) evolving according to nonlinear heterogeneous local interaction rules which is not based on Lyapunov function arguments. In particular, we focus on a class of discrete-time MASs whose global dynamics can be represented by sub-homogeneous and order-preserving nonlinear maps. This paper directly generalizes results for sub-homogeneous and order-preserving linear maps which are shown to be the counterpart to stochastic matrices thanks to nonlinear Perron-Frobenius theory. We provide sufficient conditions on the structure of local interaction rules among agents to establish convergence to a fixed point and study the consensus problem in this generalized framework as a particular case. Examples to show the effectiveness of the method are provided to corroborate the theoretical analysis.

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“…Although there are various generalizations of the classical Perron-Frobenius theorem for primitive matrices in the literature, we could not find one with the convergence statement as in Theorem 1.1, which is needed for our applications. Perron-Frobenius theory and its generalizations are relevant in many more branches of mathematics than just symbolic dynamics, including applied linear algebra, and some areas of analysis and probability theory (see for instance [AGN11], [BSS12] and [Lem06]). We expect that Theorem 1.1 will find useful applications in other contexts.…”

Section: Introductionmentioning

confidence: 99%

Perron-Frobenius theory and frequency convergence for reducible substitutions

Lustig¹,

Uyanik²

2017

Discrete &Amp; Continuous Dynamical Systems - A

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v.2 Minor revisions for submissionInternational audienceWe prove a general version of the classical Perron-Frobenius convergence property for reducible matrices. We then apply this result to reducible substitutions and use it to produce limit frequencies for factors and hence invariant measures on the associated subshift. The analogous results are well known for primitive substitutions and have found many applications, but for reducible substitutions the tools provided here were so far missing from the theory

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Nonlinear Perron-Frobenius Theory and Dynamics of Cone Maps

Cited by 6 publications

References 15 publications

Dynamics of products of nonnegative matrices

Dynamics of products of nonnegative matrices

A Nonlinear Perron-Frobenius Approach for Stability and Consensus of Discrete-Time Multi-Agent Systems

Perron-Frobenius theory and frequency convergence for reducible substitutions

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